{
  "id": "1212.6665",
  "version": "v2",
  "published": "2012-12-29T21:49:57.000Z",
  "updated": "2013-01-12T12:00:25.000Z",
  "title": "Uniform Gaussian bounds for subelliptic heat kernels and an application to the total variation flow of graphs over Carnot groups",
  "authors": [
    "Luca Capogna",
    "Giovanna Citti",
    "Maria Manfredini"
  ],
  "comment": "We have corrected a few typos and added a few more details to the proof of the Gaussian estimates",
  "journal": "Analysis and Geometry in Metric Spaces, 1 (2013) 255-275",
  "doi": "10.2478/agms-2013-0006",
  "categories": [
    "math.AP",
    "math.DG",
    "math.MG"
  ],
  "abstract": "In this paper we study heat kernels associated to a Carnot group $G$, endowed with a family of collapsing left-invariant Riemannian metrics $\\sigma_\\e$ which converge in the Gromov-Hausdorff sense to a sub-Riemannian structure on $G$ as $\\e\\to 0$. The main new contribution are Gaussian-type bounds on the heat kernel for the $\\sigma_\\e$ metrics which are stable as $\\e\\to 0$ and extend the previous time-independent estimates in \\cite{CiMa-F}. As an application we study well posedness of the total variation flow of graph surfaces over a bounded domain in $(G,\\s_\\e)$. We establish interior and boundary gradient estimates, and develop a Schauder theory which are stable as $\\e\\to 0$. As a consequence we obtain long time existence of smooth solutions of the sub-Riemannian flow ($\\e=0$), which in turn yield sub-Riemannian minimal surfaces as $t\\to \\infty$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-01-12T12:00:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35H20",
      "58J35",
      "53C44"
    ],
    "keywords": [
      "total variation flow",
      "uniform gaussian bounds",
      "subelliptic heat kernels",
      "carnot group",
      "turn yield sub-riemannian minimal surfaces"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.6665C"
    }
  }
}